Sample with size n = 100 has mean = 30. Assuming the population standard deviation is 8, construct 95% confidence interval for population mean.
To construct a confidence interval for the population mean, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / sqrt(sample size))
Step 1: Identify the necessary values:
- Sample mean (x̄) = 30
- Population standard deviation (σ) = 8
- Sample size (n) = 100
- Confidence level = 95% (which corresponds to a standard z-score of 1.96 for a two-tailed test)
Step 2: Calculate the critical value:
Since we are using a z-distribution and our confidence level is 95%, we need to find the z-score that corresponds to a 97.5% confidence level in the middle of the distribution. Using a z-table or calculator, we find that the z-score is approximately 1.96.
Step 3: Calculate the confidence interval:
Confidence Interval = 30 ± (1.96) * (8 / sqrt(100))
= 30 ± (1.96) * (8 / 10)
= 30 ± (1.96) * 0.8
= 30 ± 1.568
= [28.432, 31.568]
Therefore, the 95% confidence interval for the population mean is [28.432, 31.568]. This means we are 95% confident that the true population mean falls within this interval.